Mehringer, Josef (2015): Spectral and dynamical properties of certain quantum hamiltonians in dimension two. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

After 2004, when it was possible for the first time to isolate graphene flakes, the interest in quantum mechanics of plain systems has been intensified significantly. In graphene, that is a single layer of carbon atoms aligned in a regular hexagonal structure, the generator of dynamics near the band edge is the massless Dirac operator in dimension two. We investigate the spectrum of the two-dimensional massless Dirac operator H_D coupled to an external electro-magnetic field. More precisely, our focus lies on the characterisation of the spectrum σ(H_D) for field configurations that are generated by unbounded electric and magnetic potentials. We observe that the existence of gaps in σ(H_D) depends on the ratio V^2/B at infinity, which is a ratio of the electric potential V and the magnetic field B. In particular, a sharp bound on V^2/B is given, below which σ(H_D) is purely discrete. Further, we show that if the ratio V^2/B is unbounded at infinity, H_D has no spectral gaps for a huge class of fields B and potentials V . The latter statement leads to examples of two-dimensional massless Dirac operators with dense pure point spectrum. We extend the ideas, developed for H_D, to the classical Pauli (and the magnetic Schrödinger) operator in dimension two. It turns out that also such non-relativistic operators with a strong repulsive potential do admit criteria for spectral gaps in terms of B and V . Similarly as in the case of the Dirac operator, we show that those gaps do not occur in general if |V| is dominating B at infinity. It should be mentioned that this leads to a complete characterisation of the spectrum of certain Pauli (and Schrödinger) operators with very elementary, rotationally symmetric field configurations. Considering for the Dirac operator H_D the regime of a growing ratio V^2/B, there happens a transition from pure point to continuous spectrum. A phenomenon that is particularly interesting from the dynamical point of view. Therefore, we address in a second part of the thesis the question under which spectral conditions ballistic wave package spreading in two-dimensional Dirac systems is possible. To be more explicit, we study the following problem: Do statements on the spectral type of H_D already suffice to decide whether the time mean of the expectation value $$\frac{1}{T} \int_0^T \sps{\psi(t)}{|\bx|^2\psi(t)} \rd t$$ behaves like T^2? Here ψ(t) denotes the time evolution of a state ψ under the corresponding Dirac operator. We can answer that question affirmatively, at least for certain electro-magnetic fields with symmetry.